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Recent extensions to dynamic games of the well-known fictitious play learningprocedure in static games were proved to globally converge to stationary Nashequilibria in two important classes of dynamic games (zero-sum andidentical-interest discounted stochastic games). However, those decentralizedalgorithms need the players to know exactly the model (the transitionprobabilities and their payoffs at every stage). To overcome these strongassumptions, our paper introduces regularizations of the systems in (Leslie2020; Baudin 2022) to construct a family of new decentralized learningalgorithms which are model-free (players don't know the transitions and theirpayoffs are perturbed at every stage). Our procedures can be seen as extensionsto stochastic games of the classical smooth fictitious play learning proceduresin static games (where the players best responses are regularized, thanks to asmooth strictly concave perturbation of their payoff functions). We prove theconvergence of our family of procedures to stationary regularized Nashequilibria in zero-sum and identical-interest discounted stochastic games. Theproof uses the continuous smooth best-response dynamics counterparts, andstochastic approximation methods. When there is only one player, our problem isan instance of Reinforcement Learning and our procedures are proved to globallyconverge to the optimal stationary policy of the regularized MDP. In thatsense, they can be seen as an alternative to the well known Q-learningprocedure.

Smooth Fictitious Play in Stochastic Games with Perturbed Payoffs and Unknown Transitions